Reflectors made of linear grooves

ABSTRACT

An embodiment of a method of designing a grooved reflector comprises selecting two given wavefronts; and designing two surfaces meeting at an edge to form a groove such that the rays of each of the given wavefront become rays of a respective one of the given wavefronts after a reflection at each of the surfaces. Multiple grooves may be combined to form a mirror covering a desired area. A mirror may be manufactured according to the design.

CROSS REFERENCE TO RELATED APPLICATION

This application claims benefit from U.S. Provisional Application No.61/131,884, filed Jun. 13, 2008, which is incorporated by referenceherein in its entirety.

FIELD OF THE INVENTION

The present application relates generally to light concentration andillumination, and more particularly to reliable high efficiencyreflectors. The devices contain grooved structures that with tworeflections couple the rays of two given wavefronts exactly, with nolimitation about the groove size. The groove profiles disclosed hereinare designed with the SMS (Simultaneous Multiple Surface) method, whichis a direct method that does not require a numerical optimizationalgorithm. Additionally, the grooves vertices reside on lines which arenot restricted to be planar curves. The application of those grooves tothe substitution of metallic reflectors in known devices, as the RXI(Refraction-refleXion-Internal reflection) or XR (refleXion-Refraction)as described in U.S. Pat. No. 6,639,733 to Miñano et al.), is alsodisclosed.

In general, a “groove” as discussed in the present application is astructure with two surfaces that face partly towards each other, usuallyat least approximately in a V shape, so that a light ray can enter the Vat the open top, reflect off each side in turn, and exit through theopen top of the V. However, the present application is primarilyconcerned with devices operating by total internal reflection, so thatthe light is in a medium of higher refractive index than the mediumoutside the V shape. Typically, the inside of the groove is a dielectricmaterial such as glass or plastic, and the outside is air, so that thestructure that is optically and mathematically a groove, and isdescribed in this specification as a “groove,” is in many casesmechanically a projecting ridge on the back of a dielectric body withinwhich the “groove” is defined.

BACKGROUND OF THE INVENTION

V-shaped grooves that by two ray bounces emulate the functionality of asingle-bounce reflector of light have been proposed and used indifferent applications. For instance, grooves with flat facets andcylindrical symmetry on a flat surface are known as abrightness-enhancing film (BEF), manufactured by the 3M Corporationunder the brand name Vikuiti, for displays.

The same geometry but used in a different way (as a reflector of solarbeam radiation) was proposed to make heliostats for solar thermal energyby M. O'Neill, “Analytical and Experimental Study of Total InternalReflection Prismatic Panels for Solar Energy Concentrators” TechnicalReport No. D50000/TR 76-06, E-Systems, Inc., P.O. Box 6118, Dallas, Tex.(1976), and A. Rabl, “Prisms with total internal reflection,” SolarEnergy 19, 555-565 (1977), also U.S. Pat. No. 4,120,565 By A. Rabl andV. Rabl.

A parabolic dish reflector using V-shaped radial grooves on theparaboloid surface (i.e., the guiding lines of the grooves are containedin meridian planes of the original parabolic reflector) is beingmanufactured by the company Spectrus (http://www.spectrusinc.com/) forillumination applications. See, for instance, the “Reflexor Retrofitsystem” athttp://spectrusinc.com/products-detail/reflexor-retrofit-system-178/8/.Similar reflectors were also proposed for solar applications. See A.Rabl, Prisms with total internal reflection, Solar Energy 19, 555-565(1977) and also U.S. Pat. No. 4,120,565 to A. Rabl and V. Rabl. See alsoM. O'Neill, Analytical and Experimental Study of Total InternalReflection Prismatic Panels for Solar Energy Concentrators, TechnicalReport No. D50000/TR 76-06, E-Systems, Inc., P.O. Box 6118, Dallas, Tex.(1976). In those designs, the cross section of the grooves is flat,which limits the device performance unless the groove size is smallcompared to the receiver or source sizes.

Improving that groove-size limitation, patent application US2008/0165437 A1 by DiDomenico discloses a design method for V-shapedradial grooves the cross-sectional profile of which is not flat, andapplies that method to parabolic dish reflectors. The non-flat profileis designed by parameterizing the surface using Bezier splines and usinga numerical multi-parameter optimization method to minimize a certaincost function. The convergence of such a method to a global minimum isnot guaranteed, and paragraph [0110], page 10 of US 2008/0165437 A1specifically mentions the existence of many local minima that may trapthe optimization algorithm, and explains that the selection of the costfunction and the initial guess of the free-form surfaces are “critical.”No example of that cost function is given in US 2008/0165437 A1.

US 2008/0165437 A1 also discloses other devices as the XX (tworeflecting surfaces) or RXI whose V-shaped radial grooves are designedby such an optimization procedure. No description is given of how thenon-grooved surfaces are designed. US 2008/0165437 A1 claims inparagraph [0177], page 36 that their devices perform close to thephysical limits. However, two contradictions are visible:

1. In FIG. 6A of US 2008/0165437 A1, the edge rays at the input areclearly not transformed into edge rays at the output, which is thenecessary and sufficient condition to achieve the claimed physical limit(maximum concentration). The edge rays at the input are those containedin the surface of the cones (labeled as 305 in FIG. 3 and named so inparagraph [0146], page 33 of US 2008/0165437 A1), and the rays at theoutput must impinge on the contour of circle 610 of said FIG. 6A. Thefact the input rays are well inside the circle 610 indicates that thephase space volume is far from been fully filled, and thus it performsalso far from the concentration limit.

2. The device shown in FIG. 15B of US 2008/0165437 A1 cannot be a wellperforming concentrator device, since its thickness at the center isonly 0.19 times the diameter, which is below the compactness limit ofthat concentrator device (0.23) easily deducible from the Fermatprinciple.

Summarizing, the grooved reflectors in the prior art are limited toguiding lines which are either straight parallel lines (i.e. prismatic90° retroreflectors) or radial planar curves, and the cross sectionprofile of the groove is flat or Bezier splines optimized by numericalalgorithms.

SUMMARY OF THE INVENTION

Embodiments of the invention provide methods of designing groovedreflectors.

Embodiments of the invention provide methods of manufacturing groovedreflectors, comprising designing a reflector by a method in accordancewith an embodiment of the invention, and manufacturing a reflector inaccordance with the design.

Embodiments of the invention provide grooved reflectors designed by amethod in accordance with an embodiment of the invention, and novelgrooved reflectors per se, including reflectors identical to reflectorsthat would result from the design and manufacturing methods of theinvention, whether or not actually designed by such methods.

Embodiments of the invention provide collimators, concentrators, andother optical devices incorporating reflectors according to theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features and advantages of the presentinvention will be apparent from the following more particulardescription thereof, presented in conjunction with the followingdrawings wherein:

FIG. 1 shows a cross section of the V-shaped prismatic retro-reflector

FIG. 2 a shows one way to manufacture a linear retro-reflector is usinga dielectric prism.

FIG. 2 b is a perspective view of the same prism of FIG. 2 a

FIG. 3 is a diagram of the (p,q) plane showing the region where anincident ray (p,q,r) undergoes a single TIR (Total Internal Reflection)at each of the two sides of the V trough of FIGS. 2 a and 2 b.

FIG. 4 a shows an array of retroreflector prisms

FIG. 4 b shows the cross-section of an array of retroreflector prisms.

FIG. 5 shows the more general case of a free-form sheet.

FIG. 6 shows a radially-symmetric grooved reflector in which the grooveguiding lines are parabolas.

FIG. 7 a shows the non-planar guiding lines of another solution of thesame Cartesian oval problem as a paraboloid, in which the guiding linesintersect a pair of straight lines.

FIG. 7 b shows the grooved reflector of the solution of FIG. 7 a.

FIG. 8 shows a generic free-form conventional reflector.

FIG. 9 shows the solid angle cone of FIG. 8 represented in a plane ofφ−θ coordinates.

FIG. 10 a shows a grooved reflector designed to substitute for aconventional reflector of an XR design of revolution.

FIG. 10 b is a back side view of the grooved reflector shown in FIG. 10a.

FIG. 11 a shows a grooved reflector designed to substitute for afree-form reflector of an XR design.

FIG. 11 b is a back side view of the grooved reflector shown in FIG. 11a.

FIG. 12 shows a grooved reflector with two-level grooves generated fromguiding curves lying on two reference surfaces.

FIG. 13 shows another two-level grooved reflector showing some pointswhere groove edges generated from a first reference surface intercepton, and initiate, guiding curves on a second reference surface.

FIG. 14 shows a cross-section through an embodiment of a retro-reflectorprofile.

FIG. 15 shows a cross-section through another retro-reflector profile.

FIG. 16 a shows the ray-trace on a parabolic approximation on a designsimilar to that of FIG. 15.

FIG. 16 b shows the ray-trace on the same design as in FIG. 16 a but fora larger angular half-span of the rays of the spherical wavefront.

FIG. 17 shows an SMS2D retro-reflector profile whose function is similarto that of FIG. 15 in which convergence at the groove corner is notpossible.

FIG. 18 shows the limit case of FIG. 17 for the two points coincident atinfinity, which is the dual case of retroreflector of FIG. 1 with a realcaustic inside the groove.

FIG. 19 shows that retroreflector of FIG. 18 does not retroreflectexactly parallel rays with tilt angle α>0 as expected.

FIG. 20 a shows the first steps in the construction of the SMS2Dretroreflector with Cartesian ovals at the edges.

FIG. 20 b shows the next steps in the design of construction of FIG. 20a.

FIG. 21 shows a combination of positive lenses with essentially flatgrooves so little or no light is reflected towards the groove corners.

FIG. 22 shows an alternative design in which the grooves are concaveenough to reverse the sign of the magnification of the system.

FIG. 23 shows a retroreflector with Kohler integration by lenses on thegroove cover where there are two microlenses per groove.

FIG. 24 shows a retroreflector with Kohler integration by lenses on thegroove cover where there is one microlens per groove and the apex of themicrolens is in line with the valley of the groove.

FIG. 25 shows a retroreflector with Kohler integration by lenses on thegroove cover where there is one microlens per groove and the apex of themicrolens is in line with the apex of the groove.

FIG. 26 shows a grooved parabolic reflector similar to the one describedin FIGS. 10 a and 10 b but made of 10 free-form groove reflectors withlarge non-flat cross sectional profiles.

FIG. 27 shows the design of a large groove as a sequence of 2D designsalong the groove.

FIG. 28 shows one groove as an exact solution in 3D.

FIG. 29 shows a cross section view of a rotational-symmetric air-gap RXIdevice.

FIG. 30 shows a front surface of the device shown in FIG. 29.

FIG. 31 shows a back reflector surface and a refractive cavity surfaceof the device shown in FIG. 29.

FIG. 32 shows a cross-section view of a device similar to that shown inFIG. 29, wherein a reflector surface comprises by a V-shaped groovedreflector.

FIG. 33 shows a device similar to that shown in FIG. 29, with groovedreflectors instead of all the metallic reflecting surfaces of FIG. 29.

FIG. 34 a and FIG. 34 b show the back and front grooved reflectorsurfaces, respectively, of the device of FIG. 33.

FIG. 35 shows a close up of the front grooved reflector surface of FIG.34 b.

FIG. 36 shows a cross-section of a device that does not need the innerfront reflector of FIG. 35.

FIG. 37 shows the cross-section of a device with a lens instead of thefront reflector of FIG. 35.

FIG. 38 shows a piece of flat radial grooved film trimmed to be used foran approximately conical reflector.

FIG. 39 a shows an intensity pattern for a conventional RXI device closeto the aplanatic condition when used as a collimating source with theemitter centered on axis.

FIG. 39 b shows an intensity pattern for a conventional RXI similar tothat of FIG. 39 a when the emitter is placed off the axis.

FIG. 40 shows an intensity pattern for a grooved RXI similar to thatshown in FIG. 32 when the emitter is placed off the axis as in FIG. 39b.

FIG. 41 shows a reflecting cavity.

FIG. 42 shows a cavity similar to that of FIG. 42 but with a groovedreflector forming a reflecting cavity surface.

FIG. 43 shows a device similar to FIG. 41 but made of a singledielectric piece, in which the LED is not in optical contact with thepiece.

FIG. 44 and FIG. 45 show another possibility to provide such abrightness enhancement and collimating cavity, using a groove profile asthat of FIG. 15.

FIG. 46 shows the use of a single groove to provide such a brightnessenhancement and collimating cavity.

DETAILED DESCRIPTION 1. Introduction

A better understanding of the features and advantages of the presentinvention will be obtained by reference to the following detaileddescription of the invention and accompanying drawings, which set forthillustrative embodiments in which various principles of the inventionare utilized.

Referring to FIG. 1, an embodiment of a prismatic retro-reflector isformed by a 90 deg V shaped trough-like mirror. FIG. 1 shows a crosssection of the V-shaped reflector 11, as well as the coordinate systemused to describe it. The x-axis, perpendicular to the plane of thefigure, is the axis of translational symmetry of the linearretro-reflector. An important property of this reflector configurationis shown for an incident ray 12 with direction cosines (p,q,r),successively reflected at each side of the V trough. The incident ray 12is then reflected as ray 13 with direction cosines (p,−q,−r), that is,its components in they and z axis are retro-reflected. This has to becompared with a conventional flat reflector with its normal in the ydirection, where the direction of the reflected ray would be (p,−q,r),or compared with the cube-corner retro-reflector, where any rayreflected by all three faces leaves completely retro-reflected, i.e.,with direction (−p,−q,−r).

Let v_(i) be the incident ray vector (with components p,q,r) and letv_(o) be the reflected ray vector for a ray undergoing two reflections.The reflection law for these rays can be written:v_(o)=v_(i)−2(n·v_(i))n−2(s·v_(i))s, where n is the unit normal vectorto entry aperture 14 and s is the unit vector perpendicular both to thegroove's axis of linear symmetry and to n. In the case of FIG. 1, ncoincides with the unit vector z and s coincides with the unit vector y.The vector t is the unit tangent along the groove (x in FIG. 1). Thens·t=n·t=·n=0. This reflection law establishes that the ray vectorcomponents along the vectors n and s change in sign after reflectionwhile the remaining component, along t, keeps its sign and magnitude. Ina conventional reflection with normal vector n, the reflection law isv_(o)=v_(i)−2(n·v_(i))n and only the component of the ray vector in then direction changes its sign. The two remaining components keep theiroriginal sign and magnitude.

Since any vector v_(i) can be written in terms of its components in thetri-orthogonal system based on n, s and t asv_(i)=(n·v_(i))n+(s·v_(i))s+(t·v_(i))t, the reflection law for groovedsurfaces can be written as v_(o)+v_(i)=2(t·v_(i))t.

All the rays reaching the entry aperture 14 will undergo at least onereflection, but not all rays will have two. For instance, ray 15 leavesthe retro-reflector with a single reflection.

One way to manufacture a linear retro-reflector is using a dielectricprism, such as that shown in FIG. 2 a. FIG. 2 b is a perspective view ofthe same prism. The main interest of this solution is that thereflections 21 and 22 at the two sides of the V trough can be totalinternal reflections (TIR) which inherently have high efficiency as longas the reflecting faces are highly smooth. The entry aperture is now oneof the prism faces and consequently the rays are refracted upon enteringat 23 or exiting the dielectric prism at 24. The entering and exitingrefractions compensate for each other, so that the direction cosines ofthe reflected ray are still (p,−q,−r) as in the preceding non-dielectriccase (provided the ray undergoes two reflections). Unlike the precedingcase there are now rays such as 25 that undergo not two but threereflections, with the additional one being 26 at the entry apertureinterface. The direction of the escaping ray 27 is (p,q,−r), as in aconventional single reflection at a plane normal to the z axis.

The condition for an incident ray (p,q,r) to undergo a single TIR ateach of the two sides of the V trough is that the angle with the normalat the points of reflection 21 and 22 be greater than the critical anglearcsin(1/n) where n is the refractive index of the dielectric. Forexample if n=1.494 this condition is fulfilled by the rays (p,q,r) withthe (p,q) component being inside the solid region 31 of FIG. 3. FIG. 3shows in solid white color the region 31 of the (p,q) plane where anincident ray (p,q,r) undergoes a single TIR (Total Internal Reflection)at each of the two sides of the V trough. The cross-hatched regions 32represent values of p,q that are valid, but for which the desired doubleTIR does not occur. The condition for this double TIR is that the anglewith the normal at 21 and at 22 in FIGS. 2 a and 2 b be greater than thecritical angle arcsin(1/n) where n is the refractive index of thedielectric. By definition the direction cosines fulfill p²+q²+r²=1, sothe (p,q) component is inside the circle 32 in FIG. 3 so that p²+q²≦1.The narrowest portion of the solid region 31 corresponds to rays withp=0, i.e., rays contained in planes normal to the x axis. For these raysq must be in the range±{(n²−1)^(1/2)−1}/2^(1/2), which implies that nmust be greater than 2^(1/2)=1.414 for a non null range of q to exist atp=0 .

For rays with p≠0, i.e., for sagittal rays, two total internalreflections are achieved over a bigger range of q.

1.1 Arrays of Prisms as Reflector Surfaces

The limited range for which two TIR are achieved restricts theapplicability of these prisms as reflectors to those cases in which therays of interest impinge the prism with angular coordinates within thesolid region 31 of FIG. 3.

The grooved reflector only works for rays with angular coordinateswithin the solid region 31 of FIG. 3. In general the remaining rays passthrough the reflector at either reflection point 21 or reflection point22. This can be advantageous for solar applications because parts of thereflector can become non reflecting (and thus cannot concentrateradiation) when the reflector is not aimed at the sun, if the reflectoris aligned so that for these parts the sunlight is arriving with angularcoordinates outside region 31 of FIG. 3.

Consider an array of prisms such as that shown in FIG. 4 a, withcross-section shown in FIG. 4 b. This type of array is known as abrightness-enhancing film (BEF). Such a film is manufactured by the 3MCorporation. Unlike the prism of FIG. 2 a and FIG. 2 b, the entryaperture plane of the array 41 is located above the tips of the mirrorgrooves 42. This ensures that rays such as ray 25 of FIG. 2 a typicallydo not undergo two reflections at both sides of the same groove but onopposite sides of different grooves (besides the TIR reflection at theentry aperture, such as 26 of FIG. 2 a.

Next consider a more general case of a free-form sheet, as shown in FIG.5. The entry aperture 51 is a free-form surface with the parametricequation A=A(α,β), where α and β are parameters defining position on theentry surface 51. The unit surface-normal vector n depends on theposition of the point within the surface, i.e., n=n(α,β). The groovesare no longer linear but in this embodiment their bottom edges followcurves 52 that are located on a notional surface parallel to the entryaperture. The parametric equation of this parallel surface isA_(b)=A(α,β)−τ·n(α,β), where the constant τ is the distance 53 betweensurfaces. The equation of one of the groove guiding curves 52 can bewritten as a relationship between the parameters α and β, e.g.,φ(α,β)=0. Assume that we have chosen the parameters α and β such thatthe guiding curve equations can be written as φ(α,β)≡β−β₀=0, wheredifferent values of β₀ define different guiding curves. This means thatthe parametric equation of each curve is A_(b)=A(α,β)−τn(α,β₀), with αbeing the parameter along the curve. For the sake of brevity thisequation will hereinafter be referred to as C=C(α).

Designate as t the unit tangent to guiding curves, so that t is parallelto dC/dα, and let s be s=t×n. These three mutually perpendicular unitvectors (|s|=|t|=|n|=1) vary in direction along the guiding curve (theirdirections depend on α). Except for t they are distinct from the Frenettri-orthogonal system of the guiding curve [D. J. Struik, Lectures onClassical Differential Geometry, p. 19 (Dover, N.Y., 1988)].

1.2 Surface Equation of an Array of V-Shaped Curved Grooves

The equations of the surfaces of the groove sides corresponding to theguiding curve C(α) can be written as G₁(α,γ)=C(α)+γ(s−n) andG_(r)(α,γ)=C(α)+γ(s+n). The parameters of these surfaces are α and γ.The first equation is valid for γ<0 and defines one side of the groove.The other equation is valid for γ>0 and defines the other side. Bothequations can be unified as G(α,γ)=C(α)+γs+|γ|n, valid for any y andgiving the guiding curve when γ=0. Note that n and s depend solely on αwhile moving along a single guiding curve. The implicit surface equationof the array can be obtained as the surface that separates (bounds) thevolume defined by the union of the points belonging to the inner part(interior) of any groove. Let l(x,y,z)=0 be the implicit equation of agroove's lateral surface having a parametric equation G₁(α,γ), and letr(x,y,z)=0 be the implicit equation corresponding to the other side,G_(r)(α,γ). By properly choosing the sign of the functions l(x,y,z) andr(x,y,z), the interior of the groove can be defined by the points(x,y,z) for which the functions l(x,y,z), and r(x,y,z) are positive. Thesurface of the array of grooves (“grooved reflector”) is the boundary ofthe solid volume defined as the union of the interiors of the grooves.We will call this volume U. Let us define the “reflector aperture” asthe surface of the convex hull(http://en.wikipedia.org/wiki/Convex_hull) of the complementary volumeof U. The intersections of a groove surface's sides with the reflectoraperture of the groove array are the “groove aperture boundary” curves54.

When the groove-array surface forms one face of a sheet, the other faceis called the “sheet aperture” 51. For this solid to be a single piece,in the case of the V-shaped grooves, it is necessary for the adjacentguiding curves to be closer than 2τ, where τ is the grooved reflectorthickness, defined as the maximum distance between any point of theguiding curves 52 and the sheet aperture 51. (This distance is measuredalong the local normal to the sheet aperture.) The equation of the sheetaperture, together with the grooved surface itself, completely definesthe solid sheet.

1.3 Surface Equation of an Array of Curved Grooves with ConstantCross-Section

Consider next the case of non V-shaped grooves, which are also ofinterest for optical applications. A cross-section curve is contained inthe corresponding plane transverse to the guiding curve C(α). Ofgreatest interest are grooves with identical cross-sections, because thegrooved surfaces are then easier to tool than in the case of a varyingcross-section. If the cross-section is constant along the groove, thenevery cross-section curve can be defined with the same parametricequation J=J(x,y,γ) when expressed as a function of two unit vectors xand y contained in the transverse plane. The parameter along thecross-section curve is γ. For the V-shaped cross-section case,J(x,y,γ)=γx+|γ|y. In the general case, the equation of the groovesurface is G(α,γ)=C(α)+J(s,n,θ(γ),γ), where s and n were previouslydefined with directions varying along the guiding curve (i.e., varyingwith α) and θ(γ), the angle between the vector x (or y) and s (or n), isa function of the parameter γ along the curve trajectory.

1.4 Thin Reflector Approximation

Assume that the cross-sectional sizes of the grooves and the curvaturesof the guiding curves are small enough to be locally linear, so thatwhen a ray exits through the aperture of the array of grooves it does soat the same point of the aperture where it enters, and so that thedeflection suffered by the rays is the same as being in a linearsymmetric groove with axis of linear symmetry being tangent to theguiding curve. This situation is known as the thin reflectorapproximation. Note that there is no assumption about the curvatures ofthe cross section curves. The size that makes this approximation validdepends upon the application. When the groove is V-shaped the thinreflector approximation means that the rays are reflected at the pointof incidence on the reflector aperture, and most of them (those raysundergoing two reflections in the groove) satisfy the reflection lawv_(r)+v_(i)=2(t·v_(i))t.

2. The Cartesian Oval Problem Design with V-Shaped Retro-ReflectorArrays

The problem is then to design a free-form grooved sheet that reflects agiven incident ray vector field v_(i)(r) into another known exiting rayvector field v_(o)(r), with r being a point of the space, i.e.,r=(x,y,z). The problem of finding a refractive or reflective surfacethat transforms any vector field v_(i)(r) into another ray vector fieldv_(o)(r) is called the Generalized Cartesian Oval problem [R. Winston,J. C. Miñano, P. Benítez, Nonimaging Optics, (Elsevier, 2005), seeespecially p. 185].

The reflection law for an array of V-shaped grooves establishes withinthe thin sheet approximation that the tangent to the grooves' guidinglines is parallel to v_(o)+v_(i), so that it fulfills (v_(o)+v_(i))×t=0.The only condition on n (besides being a unit vector) is that it benormal to t. Let ψ(r)=0 be the (implicit) equation of the entry aperturesurface of the retro-reflector sheet. Then its gradient ∇ψ must beparallel to n at the points of the surface ψ(r)=0. Then∇ψ·{v_(o)(r)+v_(i)(r)}=0. This is a first order linear differentialequation with the function ψ(r) as the unknown. Note that the vectorfields v_(i)(r) and v_(o)(r) are known because they have beenprescribed. The integration of this equation, together with the boundaryconditions, gives the desired surface ψ(r)=0. A suitable boundarycondition is an arbitrary curve to be contained in the surface ψ(r)=0.This is remarkably different from the Generalized Cartesian Ovalproblem, where the differential equation for the conventional reflectorsurface ψ_(c)=0 that reflects the ray vector field v_(i)(r) into the rayvector field v_(o)(r), establishes that the normal to the surface mustbe parallel to v_(o)(r)−v_(i)(r), i.e., ∇ψ_(c)×{v_(o)(r)−v_(i)(r)})0.This is a total differential equation, so that only a point on thesurface can be a boundary condition. The extra degree of freedom in theconventional reflector design problem is due to the extra freedom inchoosing the guiding curves once the aperture surface is given.

Once ∇ψ·{v_(o)(r)+v_(i)(r)}=0 has been integrated (so that ψ(r)=0 isknown), it only remains to calculate the guiding curves and to select τ.A guiding curves is calculated by integration of the vector fieldv_(o)(r)+v_(i)(r), initiated from any point contained in ψ(r)=0. Thesecurves are necessarily contained in ψ(r)=0 because this surface has beenobtained with the condition of being tangent to the vector fieldv_(o)(r)+v_(i)(r). The initial points for integration are selected tofulfill the thin sheet approximation. The angle θ(γ) is chosen tomaximize the number of rays deflected according to the lawv_(r)+v_(i)=2(t·v_(i))t (i.e., the number of rays undergoing tworeflections) among the rays of interest. In general this conditionimplies that the V-shape local plane of symmetry is tangent to n, (i.e.,tangent to ∇ψ) or tangent to v_(r)−v_(i). The sheet thickness τ isselected for mechanical stiffness.

As an example consider v_(i)(r)=r/|r|=(x,y,z)/(x²+y²+z²)^(1/2) andv_(o)(r)=z, where z=(0,0,1) in Cartesian coordinates. The solution isany surface containing a family of guiding curves that are parabolaswith focus at the origin and axis z. The solution shown in FIG. 6 is aparaboloid 61. In this solution the boundary condition is a circle 62normal to the z axis and centered on the z axis. The guiding curves 63are radial and the planes of symmetry of the grooves coincide with themeridian planes. In this case, the solution coincides with theconventional reflector case mentioned above.

FIG. 7 a shows the solution 71 when the boundary condition is that thesurface must contain the straight line 72. The guiding curves 73 arecontained in meridian planes (planes containing the z axis). FIG. 7 bshows the grooved reflector of this solution, with the local planes ofsymmetry of the grooves being tangent to the surface normals.

This design procedure does not ensure that the rays of the vector fieldv_(i)(r) that impinge on the surface will undergo two TIRs at thegrooved reflector. This condition should be checked after the surfacehas been calculated. Some points of the surface will fulfill it andothers will not.

Retro-reflector arrays that lie on developable surfaces are ofparticular practical interest because they can be molded on flat sheets,the same as BEF film, and tailored to the application by trimming andbending.

3. Design of Free-Form Retro-Reflectors Arrays with ConstantCross-Section Grooves

Another design problem is a free-form grooved reflector that replaces afree form conventional reflector in a nonimaging application. Assumethat in this application rays from a source are to be sent to a target.

3.1 Calculation of the Guiding Curves

There follow three procedures to obtain guiding lines.

1) FIG. 8 shows a generic free-form conventional reflector 81 that isassumed to be a part of a nonimaging system transferring optical powerfrom a source to a target. The rays linking source and target that hit ageneric point 82 are represented by the solid angle characterized by thespherical sector 83. This solid angle represents the directions of theserays before being deflected at the point 82. The surface normal at thepoint 82 is n. Consider the set of planes containing the vector n normalto the surface 81 and look for that plane that divides the solid angleinto the most highly symmetric parts.

For this purpose, consider a set of spherical coordinates ρ,φ,θ centeredat the point 82 and such that the z axis coincides with the direction ofn. The solid angle can be represented as the cross-hatched region 92 ina chart of φ−θ coordinates 91, such as the one shown in FIG. 9. Thesymmetric image of 92, with respect to a plane 93 tilted an angle φwithrespect to the reference, is the region 94. We look for the plane 93maximizing the intersection region between 92 and 94. When the region 92is perfectly symmetric with respect to some plane 93, it can be exactlyoverlapped with region 94 for some angle φ. Once this maximizing problemhas been solved the vector t can be calculated at any point 82 as theunit vector tangent to the free-form surface 81 at 82 and contained inthe maximizing plane 93. This procedure enables the calculation of avector field on the surface 81 and its integration into a family ofcurves that will be the guiding curves of the grooved reflector.

2) In a 3D SMS (Simultaneous Multiple Surface) design (as described inU.S. Pat. No. 7,460,985 to Benitez et al.) there is a simplifiedprocedure to calculate guiding curves. In this design procedure, twonormal congruences of rays are used to design the optical surfaces oncethe “seed rib” on one of the surfaces is known (the “seed rib” is acurve on a surface, along with the surface normals at the points of thecurve). A normal congruence of rays is a set of rays for which there isa family of surfaces normal to their trajectories, i.e., the wavefronts.Assume that s₁(x,y,z) and s₂(x,y,z) are the optical path lengths alongeach of these congruences, i.e., the equations s₁(x,y,z)=constant definethe various wavefronts. Then, for most cases of interest, the guidinglines are the intersection of the mirror surface M with the family ofsurfaces defined by the equation s₁(x,y,z)−s₂(x,y,z)=constant.

3) As mentioned above in Section 2, in a Generalized Cartesian ovaldesign the guiding curves can be calculated as the integral lines of thevector field v_(o)(r)+v_(i)(r).

The choice of which of these guiding conditions to use depends upon theprocedure used to design the entire optical system, as well as theapplication.

3.2 Design of the Groove

The angle θ(γ) is selected depending on the particular groovecross-section. In the case of a V-shaped cross-section this angle ischosen so the groove's local plane of symmetry is the aforementionedmaximizing plane. The guiding curves C(α) together with the groove'scross-section J(s,n,θ(γ),γ), n and θ(γ) completely characterize thegrooved reflector.

Consider the case when the grooved reflector is one face of a dielectricsheet with its other face being a surface parallel to the genericfree-form surface 81. Assume that the sheet is thin enough that thisparallel surface is effectively identical to the surface 81. In thiscase, the rays of the type 25 of FIG. 2 a that do not follow the generallaw are not lost in a nonimaging application since they are stillreflected within the bundle of rays connecting source and target. Thesame thing happens to the light undergoing Fresnel reflection at theparallel surface, i.e., this light is not lost because it follows thesame trajectory as when 81 is a conventional reflector. For theremaining rays that are reflected by TIR, this grooved reflector sheetis potentially very efficient. Nevertheless there are still an importantsource of optical losses: first there are two reflections instead of theone of a conventional reflector. This means necessarily more sensitivityto surface roughness and surface errors. Second, the sheet thicknesscannot be arbitrarily small. Selecting the sheet thickness must takeinto account the fact that the spacing of the grooves is related to it,so that the small thickness of the thin reflector approximation impliessmall groove spacing and thus more grooves per unit of surface, with alonger total groove length per unit of entry aperture surface. Thecorners (edges) of the grooves cannot be geometrically perfect, becausethe minimum radius achievable is non-zero, so there are optical lossesthe magnitude of which depends on the total length of the corners. Thus,an increased number of grooves means an increased number of corners andan increase of optical losses due to this effect. Moreover the spacingbetween the tips of the grooves and the entry aperture also affects thethin sheet approximation, and a ray impinging in a groove can bereflected towards another neighbor groove (for instance, rays such as 25of FIG. 2 a.

This reflector sheet can be as robust as the dielectric material ofwhich it is made.

FIG. 10 a shows the grooved reflector 101 designed to substitute for aconventional reflector of an XR design of revolution as described inU.S. Pat. No. 6,639,733 to Miñano et al. and intended for solarphotovoltaic applications. This conventional reflector is close to aparabolic shape (though not exactly parabolic). The entry aperture 102is smooth. The V shaped grooves 103 are on the back side.

FIG. 10 b is a view of the back side view of the same grooved reflector,showing that the guiding curves lie in meridian planes.

FIG. 11 a shows the grooved reflector 111 designed to substitute for afree-form reflector of an XR design or of an XX (two reflections)design, such as those shown in U.S. Pat. No. 7,460,985 to Benitez etal., titled “Three-Dimensional Simultaneous Multiple-Surface Method andFree-form Illumination-Optics Designed Therefrom”, see also U.S. patentapplication Ser. No. 12/075,830 filed Mar. 14, 2008 (publication no.2008-0223443 A1) titled “Optical Concentrator, especially for SolarPhotovoltaics”. In this case, the guiding lines 112 are not necessarilycontained in planes. The V shaped grooves 113 are on the back side andcan be seen more clearly in FIG. 11 b.

4. Guiding Curves Lying on Different Surfaces

The distance between adjacent guiding curves may vary along the curves.For instance, in a rotationally symmetric system with radial grooves,the guiding curves diverge when moving away from the axis. Consequentlythe size of the groove cross-section increases. This can be undesirablefor several reasons: (a) the density of grooves is a trade off-betweenlosses due to the rounding on the corners and the thin reflectorapproximation for those designs based on this approximation; (b) if thereflector is going to be made by plastic injection then it is desirableto have constant sheet thickness. To avoid this change of groovedensity, new grooves can be inserted between the original grooves alongthe parts of the original grooves that are relatively widely spaced. Theends of the new grooves can be an important source of losses if the newgrooves are not generated properly. Next it is shown how to create newguiding curves with grooves intersecting with the old ones (and thusincreasing the density of grooves), while simultaneously generating nonew losses.

Consider replacing a conventional reflector M by a grooved one. Choose anormal congruence of rays C that represents the rays of interest in aparticular application. For instance, in an LED collimator design therays of interest might be taken to be those issuing from the centralpoint of the LED chip.

Designate as C_(b) and C_(a) the set of rays before and after thereflection on the mirror M. The mirror transforms the set of rays C_(b)into C_(a), or, to restate, the mirror M is a reflective solution of theGeneralized Cartesian oval problem of coupling the sets of rays C_(b)and C_(a). It is not the only solution, however. Different optical pathlengths from C_(b) to C_(a) give different mirrors that are akin toparallel sources.

Let M₁ be one of these other mirrors (it is preferable to choose onenear M). Now substitute M₁ for the original mirror M and trace the raysof interest through the new system and calculate new guiding curves onM₁ as explained in Section 3.1 above. This procedure can be repeated toobtain guiding curves on different reflective solutions M_(i) of theGeneralized Cartesian oval problem.

Now create the grooves with the guiding curves on M, as explained inSection 3.2. In the regions where the guiding curves on M diverge, thecurves intersecting adjacent grooves (the groove edges) will separatefrom M. With sufficient separation, these curves will intercept M₁. Nextcalculate the points where these curves intercept the surface M₁ andtake the guiding curves on M₁ that pass through these points. Create thecorresponding grooves and intersect them with the ones generated from M.

FIG. 12 shows the grooved reflector 121 with grooves generated fromguiding curves 122 laying on M and grooves with guiding curves 123laying on M₁. FIG. 13 shows another two-level grooved reflector 131showing some points 132 where the groove edges generated from Mintercept the surface M₁ and consequently initiates the guiding curveson M₁.

5. Types of Groove Cross-Sections

The 90° V-shaped groove is just one possible profile. In general thisprofile only works well when the groove cross section is small comparedwith the size of the source or the target.

Further strategies follow to define groove cross-sections:

5.1 SMS 2D Designs with Series Expansion at Vertex

Two different SMS 2D design problems are considered next. The first one(Type I) is stated as follows: Two wavefronts in 2D geometry are givensuch that there is an associated ray common to both (i.e., the raytrajectory that is perpendicular to both wavefronts), with a point alongthat ray trajectory also specified. In this section, the design problemis to design two mirrors such that the rays of one of the wavefrontsbecome rays of the other wavefront after a reflection at each of thesurfaces (either one first). The two mirrors meet at the prescribedpoint. This design will have direct application to the design of theCartesian oval type grooved reflectors described above.

The second design problem (Type II) is stated as follows: Two wavefrontsin 2D geometry and a point are prescribed. The point is such that therays of the specified wavefronts do not coincide as they pass throughthat point. Two mirrors must be designed such that the rays of any ofthe wavefronts become reverse rays of the same wavefront aftersuccessive reflections by each of the surfaces. The two mirrors meet atthe prescribed point. This design will have direct application for thedesign of cavities discussed in Section 9, “CAVITIES WITH GROOVEDREFLECTORS” below.

Both design problems (Types I and II) are closely related when the SMSmethod is used. Type II is first described in detail, and Type I isdescribed by highlighting particular changes to the procedure for TypeII.

FIG. 14 shows an example of a Type II design problem, in which the twowavefronts are any two circles with centers at the points 142 and 143,for instance the circles of radius zero. The specified point is 141,which fulfills the requirement that it is not aligned with wavefronts142 and 143. If the problem was to design a single reflector, thesolution would be the canonical Cartesian oval problem: its solution isthe ellipse 144 with foci at 142 and 143. In the present case, however,two deflections are required.

Consider the rays 145 and 146 passing through the point 141 andbelonging one to each wavefront 142, 143. Divide each wavefront into twosets at the point where the rays 145 or 146 hit the wavefront. Moreexactly, divide each bundle into rays defined by each wavefront, toobtain bundles 147, 148, 149 and 150. Designate as input bundles the twobundles the rays of which pass to the right of point 141 (i.e., bundle147 from wavefront 142 and bundle 148 from wavefront 143) and designatethe remaining ones as output bundles, (i.e., bundle 150 from wavefront142 and bundle 149 from wavefront 143).

Consider an SMS 2D system with 147 and 148 as input bundles, 149 and 150as output bundles, the bundle 147 being transformed into 150, and bundle148 into 149 after the two reflections, selecting the optical pathlengths such that the two mirrors pass through 141. FIG. 14 shows anexample. The design procedure is as follows:

1). Calculate the optical path length from each input bundle to itscorresponding output bundle. This is simple since we know that bothdeflecting curves (both mirrors in the example) meet at the point 141,where both reflections occur for the rays 145 and 146 reaching it. Forinstance the optical path length from a wavefront of bundle 147 (takethe point 142 as this wavefront) to a wavefront of the bundle 150 (takethe same point as wavefront) is simply twice the optical path lengthfrom 142 to 141.

2). Calculate the normal vector of the mirrors at point 141. Note thatsince the bundle 147 is transformed into 150, this means that the raycoming from 142 must be reflected back to the same point. A similarresult obtains for point 143. For this to be fulfilled at the point 141it is necessary that the normal vectors to the two mirrors be orthogonal(as in a conventional linear retro-reflector). This is simple to prove:let v₁₄₂ be the unit vector of the ray coming from point 142 to 141, andlet n₁ and n₂ be the two normal vectors to the mirrors at 141. The rayvector after reflection by either surface must coincide, i.e.,v₁₄₂−2(n₁·v₁₄₂)n₁=−v₁₄₂+2(n₂·v₁₄₂)n₂ (the reflection law has been used).This equation can be re-written as v₁₄₂=(n₁·v₁₄₂)n₁+(n₂·v₁₄₂)n₂ whichcompared with the decomposition of the vector v₁₄₂ in terms of thevectors n₁ and n₂ reveals that the only condition to fulfill is that n₁and n₂ be orthogonal. Fortunately, this is the same condition neededwhen considering the bundles 148 and 149 and the ray unit vector v₁₄₃coming from 143. Then one of the normal vectors can be chosen, forinstance n₁, thus fixing the other. In general n₁ and n₂ are chosen tomaximize the power that is reflected by TIR and to minimize losses dueto the finite size of the grooves, which results in (n₁+n₂) beingparallel to (v₁₄₂+v₁₄₃).

3). Starting the SMS 2D procedure requires a point on one of the twomirrors and the normal of the mirror at that point. Although the point141 belongs to both mirrors, it is unfortunately a point of convergenceso that the SMS procedure cannot progress from there, and a furtherpoint is required. It can be obtained by considering the solution thatadmits a series expansion of one of the mirrors at the point 141 (sothat the mathematical function defining the profile is said to beanalytic around point 141). From the practical point of view it sufficesto obtain the neighboring point 141 a by means of a linear approximationof the mirror profile near the point 141. This other point 141 a can beas close as necessary to 141, i.e., we can choose it close enough so thelinear expansion is a good approximation of the mirror profile.

4). Reflect the ray from 143 on starting point 141 a and calculate thepoint 141 b along that reflected ray that matches the optical pathlength again towards 143 after the second reflection at 141 b. Computethe normal vector at 141 b consistent with the ray trajectory from 141 ato 141 b to 143.

5). Analogously, reflect the ray from 142 at 141 b and calculate thepoint 141 c along that reflected ray that matches the optical pathlength again towards 142 after the second reflection at 141 c. Computethe normal vector at 141 c consistent with the ray trajectory from 141 bto 141 c to 142.

6). Repeat the calculations of steps 4) and 5), using 141 c as astarting point and so on, to obtain sequences of points on both mirrorsthat are separate from point 141.

7). Interpolate a smooth curve between points 141 a and 141 c, with theadditional condition that the normal vector to that curve at the edgescoincides with the normal vectors at 141 a and 141 c. Since the point141 a was selected very close to 141, the distances from 141 to 141 band from 141 a to 141 c can also be expected to be small, so that theinterpolating curve can have a small arc length relative to thepractical facet size.

8). Repeat the calculations of steps 4), 5) and 6) using the points ofthe interpolating curve as a starting point. This calculation willprovide the intermediate points between the points of the sequencesstarted at 141 a and 141 b.

Regarding the Type I design problem described above, FIG. 15 shows anexample in which the two wavefronts are any two circles with centers atthe points 152 and 153, for instance the circles of radius zero. The raycommon to both wavefronts has trajectory 155, and the specified point is151. The Type I design problem is to design two mirrors such that therays of one of the wavefronts become rays of the other wavefront aftertwo reflections by each one of the surfaces (regardless of order). Thetwo mirrors meet at the prescribed point.

The design procedure is essentially the same as described for Type II.Divide each bundle of rays associated with the wavefronts into two sets,bounded by the common ray 155. thus ray bundles 157, 158, 159 and 156are obtained to play the role of bundles 147, 148, 149 and 150,respectively, in the Type II problem described in FIG. 14. Similarly,points 151 a, 151 b and 151 c are analogous to 141 a, 141 b and 141 c.

Two limit cases of both SMS design problems Type I and II areremarkable. First, when points 152 and 153 converge with each other to asingle point and the normal vectors at 151 are symmetric with respect tothe ray 155, the SMS 2D calculation converges to two confocal symmetricparabolas passing through 151 and with their axes perpendicular to thestraight line 151-152. These confocal symmetric parabolas have beensuggested as a retroreflector for instance, by Ralf Leutz, Ling Fu, andHarald Ries, “Carambola optics for recycling of light”, Applied Optics,Vol. 45, Issue 12, pp. 2572-2575. Second, when points 142 and 143(analogously 152 and 153) are taken to infinity along the samedirection, the former SMS 2D calculations leads at that limit to theconventional right angle flat-facet retroreflector discussed above.Therefore, in these particular cases flat profiles and parabolicprofiles turn out to be exact solutions of the SMS 2D design problem,with analytic profiles around point 141.

5.2 Parabolic Cross Sections

In general, the SMS 2D design of the previous section leads to asphericprofiles. If the facet is small enough, however, the profiles can beapproximated by a low order truncation of the series expansion. Thefirst-order approximation is just the right-angle flat-facet cornerdiscussed above. The second order approximation means that the mirrorsare approximated by parabolas (or, alternatively, by circumferences) andbigger facets than the flat-facet profiles can be used for the sameoptical performance.

This second-order approximation can be obtained calculating the mirrorcurvature radii (or equivalently, the second derivatives of theprofiles), at the point 141 for Type II designs and point 151 for TypeI. The relationship between the radius of curvature ρ_(i) of an incidentwavefront, the radius of curvature ρ_(r) of the reflected wavefront, andthe radius of curvature ρ_(m) of the mirror at the point of reflectionis cos(α)(1/ρ_(i)−1/ρ_(r))=2/ρ_(m), where α is the incidence anglebetween the normal to the surface and the reflected ray. The wavefrontcurvatures are taken as positive if the bundle is diverging and themirror curvature is positive when the mirror is convex. Applying thisrelationship to the bundles 147 148, 149 and 150 at point 141 gives asolvable four-equation linear system the unknowns of which are theinverse of the radii of curvature of the two mirrors at the same point141, and the inverse of the radius of curvature of the bundles betweenboth reflections. Note that the absolute value of the radius ofcurvature of the bundle 147 is equal to that of 150. The same thinghappens for the bundles 148 and 149.

As an example, consider the case of the Type II design in which points141, 142 and 143 are collinear and 143 is placed at infinity. If thenormal vectors at 141 are symmetric with respect to the straight linejoining 141 and 142 (so α=π/4), the SMS solution will be symmetric withrespect to that line, so that the second-order approximation will work.Therefore, the four linear equations reduced to these two:

$\begin{matrix}{{{\left( \frac{1}{\sqrt{2}} \right)\left( {\frac{1}{\rho_{i}} - \frac{1}{\rho_{int}}} \right)} = \frac{2}{\rho_{m}}}\;{{\left( \frac{1}{\sqrt{2}} \right)\left( \frac{1}{\rho_{int}} \right)} = \frac{2}{\rho_{m}}}} & (1)\end{matrix}$where ρ_(i) coincides with the distance between 141 and 142, so thatρ_(m)=√{square root over (32)}ρ_(i).

The equation of the approximating parabola is thus given by:

$\begin{matrix}{z = {{{{\tan(\alpha)}y} + {\frac{\left( {1 + {\tan^{2}(\alpha)}} \right)^{3/2}}{2\rho_{m}}y^{2}}} = {y + {\frac{1}{4\rho_{i}}y^{2}}}}} & (2)\end{matrix}$

FIG. 16 a shows a ray trace of this design. Note that in this design thesymmetric parabolas are not confocal, and nor is the focus of either ofthem at the primary focal point 161 a. The parabolic approximation isgood in this example if the half-angle 162 a subtended by theretroreflector from the point 161 a is about 5° full angle or less. Fora larger angle 162 b (see FIG. 16 b) the rays no longer converge ontothe small-angle focal point 161 b, instead forming a caustic. For thesegreater angles, a higher-order polynomial approximation or the exact SMS2D profile should be used instead.

5.3 Designs without Vertex Convergence

In the previous designs, a ray impinging on one mirror infinitesimallyclose to the vertex is secondly reflected on the other mirror similarlyclose to the vertex. However, we can also build other families ofsolutions in which that condition does not apply.

In the framework of the SMS 2D designs discussed before, let us considera third SMS 2D design problem (Type III) which is stated as follows: Twowavefronts in 2D geometry are given and a point is given such that thetwo rays associated to the wavefronts passing through that point are notcoincident. Our design problem is to design two mirrors that meet at theprescribed point such a point so the rays of one of the wavefrontsbecome rays of the other wavefront after reflections in the two surfacesin either order. FIG. 17 shows an example of this type of problem inwhich the prescribed point is 171 and the two wavefronts are sphericalcentered at point 172 and 173. The sequence of points calculated by theSMS is in this case 171 a, 171 b, 171 c and 171 d.

The difference between this design problem Type III and the problem TypeI described before is that in design Type I the two rays associated tothe wavefronts passing through the meeting point of the reflector werecoincident. That condition led to the result in Type I that the slopesof the two mirrors at that point form a 90° reflective corner (no matterhow it is oriented), and the two reflections in that corner willtransform the rays as desired. However, the condition of non-coincidenceof the two rays at the corner in Type III leads to the result that thereis no corner that produces the required ray-transformation. However,this does not prevent the SMS method from being applied.

Analogously to cases I and II, there are also two limit cases of TypeIII SMS design problems that are remarkable. In the first of thosecases, points 172 and 173 coincide, and the normal vectors of thereflectors at 171 are symmetric with respect to the ray linking 171 and172. The particular case where such a point 172 (173) is located atinfinity is shown in FIG. 18. The groove shown in FIG. 18 is calculatedwith the condition that a ray impinging vertically downwards on thepoint with coordinate x is, after the two reflections on the groove,sent back vertically at coordinate x+a, where a is a given quantity.This is in fact a dual of the groove of flat facets of FIG. 1 for whicha ray impinging vertically downwards at the point with coordinate x is,after the two reflections on the groove, sent back vertically atcoordinate a−x (where a is in this case the x coordinate of the vertexof the flat groove). That family corresponds to two mirrors in whichthere is a solution at the vertex. In this solution (as in that of FIG.17) the reflected rays will cross inside the groove, forming a real(i.e., non-virtual) caustic.

The optical path length condition says:−y _(R)+√{square root over ((2a)²+(y _(R) −y _(L))²)}{square root over((2a)²+(y _(R) −y _(L))²)}−y _(L)=2a  (3)

So:

$\begin{matrix}{{\left( {2a} \right)^{2} + \left( {y_{R} - y_{L}} \right)^{2}} = \left( {{2a} + \left( {y_{R} + y_{L}} \right)} \right)^{2}} & (3) \\{{{- 2}y_{R}y_{L}} = {{4{a\left( {y_{R} + y_{L}} \right)}} + {2y_{R}y_{L}}}} & (4) \\{{{y_{R}y_{L}} + {a\left( {y_{R} + y_{L}} \right)}} = 0} & (5) \\{y_{L} = \frac{- {ay}_{R}}{a + y_{R}}} & (7)\end{matrix}$

The reflection law at B:

$\begin{matrix}{{\left( {0,1} \right) \cdot \left( {1,\frac{\mathbb{d}y_{R}}{\mathbb{d}x}} \right)} = {\frac{\left( {{2a},{y_{R} - y_{L}}} \right)}{\sqrt{\left( {2a} \right)^{2} + \left( {y_{R} - y_{L}} \right)^{2}}} \cdot \left( {1,\frac{\mathbb{d}y_{R}}{\mathbb{d}x}} \right)}} & (8)\end{matrix}$

Using Equation (3) in Equation (8):

$\begin{matrix}{{\frac{\mathbb{d}y_{R}}{\mathbb{d}x}\left( {{2a} + y_{R} + y_{L}} \right)} = {{2a} + {\left( {y_{R} - y_{L}} \right)\frac{\mathbb{d}y_{R}}{\mathbb{d}x}}}} & (6) \\{\frac{\mathbb{d}y_{R}}{\mathbb{d}x} = \frac{a}{a + y_{L}}} & (7)\end{matrix}$

And substituting Equation (7):

$\begin{matrix}{\frac{\mathbb{d}y_{R}}{\mathbb{d}x} = {1 + \frac{y_{R}}{a}}} & (8) \\{\frac{\mathbb{d}y_{R}}{a + y_{R}} = \frac{\mathbb{d}x}{a}} & (9) \\{{L\;{n\left( {a + y_{R}} \right)}} = {\frac{x}{a} + b}} & (10)\end{matrix}$

Since y_(R)=0 for x=a:

$\begin{matrix}{b = {{L\;{n(a)}} - 1}} & (11) \\{{y_{R} + a} = {a\;{\mathbb{e}}^{({\frac{x}{a} - 1})}}} & (12)\end{matrix}$

Thus by Equation (7):

$\begin{matrix}{{y_{L} + a} = {a\;{\mathbb{e}}^{- {({\frac{x}{a} + 1})}}}} & (13)\end{matrix}$

The height of the groove is:

$\begin{matrix}{{{y_{R}\left( {x = {2a}} \right)} - {y_{R}\left( {x = 0} \right)}} = {{{- a} + {a\; e} - \left( {{- a} + \frac{a}{e}} \right)} = {a\left( {e - \frac{1}{e}} \right)}}} & (14)\end{matrix}$

Better coordinates are obtained by translation and scaling so:a=eX=xY=y+a  (15)

So the full profile is given by:Y=e ^(|X|1e)  (16)whose aperture is 4e and its height is e²−1.

The full angle at the bottom is 2a tan(1/e)=139.6°, to be compared withthe 90° of the normal inverting retroreflector. In an array, the fullangle at the top is then 2a tan(e)=40.4°, to be compared with the 90° ofthe normal inverting retroreflector.

FIG. 19 shows that this retroreflector does not retroreflect exactlyparallel rays with tilt angle α>0 as expected, since it is designed asaplanatic for α=0 (i.e., 172 and 173 coincident at infinity). To exactlyretroreflect the rays of α>0, the SMS design method Type III of FIG. 17is applied with 172 and 173 not coincident at infinity (separated by thedesign angle 2α).

5.4 SMS 2D Design with Cartesian Ovals at the Rim

As mentioned before, Type I and Type III SMS designs can be done toretro-reflect the rays impinging on the groove with incident angles±αwith respect to the symmetry line of the groove (Type I shown in FIG.14, and Type III in FIG. 17, where 142, 143, 172 and 173) must belocated at infinity). As a consequence of the Edge Ray Theorem ofNonimaging Optics, the rays inside±α will be retroflected also inside±α.However, Type I and Type III SMS designs will not transform all the raysimpinging within angles±α onto themselves, because some rays will missthe reflection at the edges. In order to avoid such losses, the SMS2Ddesign method can be applied for the groove design with the edge rayassignment shown in FIG. 20 a and FIG. 20 b. This problem can be calledthe Type IV problem. In this case, not only rays tilted±α (as 208 and2011) play a role in the design, but also the ray fans passing throughthe groove edges 201 and 202. This design is started at the groove edges201 and 202 building the two Cartesian ovals 2010 and 206 (parabolas inthis case in which the wavefronts are generated by points at infinity)that focus the ray fans passing reflected at the groove edges 201 and202 towards the rays of the prescribed wavefronts (parallel to 208 andto 2011, respectively, in this case). Rays 204 and 203 are mutuallyretroreflected, and edge rays 209 and 205 mark the rims 2012 and 207 ofthe Cartesian ovals. FIG. 20 b shown the next steps in the calculationof the groove profile. The new portion 2017 is calculated with thecondition that the rays 2013 (tilted+α) reflected on the known portion2010 are transformed into rays 2016 (tilted−α). Analogously, new portion2018 is calculated with rays 2015 and 2014. The process is repeated,advancing the profiles until they come close. Convergence, although notneeded in practice (manufacturing the tip will need some radius) mayoccur but the profiles will not be analytic at the convergence point.

5.5 Kohler Integration

The Kohler integration is done with the two mirrors of the groove in thesame way as disclosed in Patent Application No. WO 2007/016363 A2 byMiñano et al. The profiles will be composed by sections of Cartesianoval pairs forming a fractal-type groove. For the example of the pointsat infinity, the Cartesian oval pairs will be approximately parabolasthat will scale down when getting closer to the groove corner.

6. Combination of Grooves and Lenses

6.1 Minimizing Vertex Rounding Losses

The finite radius of the tips of the grooves when manufactured causesray losses. In order to minimize this negative rounding effect, FIG. 21discloses the combination of positive lenses with essentially flatgrooves so no light is reflected towards the groove corners. Since theangle of the lens joints is much greater than the angles of the grooves,the finite radius losses introduced by the lens joints are much smaller.

FIG. 22 discloses an alternative design in which the grooves are convextowards the light side (concave towards the mechanical outside) enoughto reverse the sign of the magnification of the system. The profiles ofthese groove designs can be calculated with the methods describedbefore.

6.2 Kohler Integration

The Kohler integration can be done with the lens added on the groovecover. There are three particular cases with special interest. The firstone is shown in FIG. 23, where there are two microlenses per groove. Thesecond one is shown in FIG. 24, and in this case there is one microlensper groove and the apex of the microlens is in line with the valley ofthe groove while in the other case (FIG. 25), the apex of the microlensis in line with the apex of the groove. The acceptance angle of theincoming radiation for which TIR is achieved in both reflections isincreased in a configuration of the first type. The configuration shownin FIG. 25 is more interesting than the one in FIG. 24 to decrease thelosses due to the rounding of the corners. In plastic moldingtechniques, the corners are not perfectly reproduced. There is arounding effect. The corners get a minimum radius. If the corner ismechanically convex (the angle inside the dielectric material is lessthan 180 deg) the minimum radius with current commercial plastic moldingtechniques is in the range of 15 microns. If the angle is concave theminimum radius is smaller (about 5 microns). The FIG. 25 configurationavoids sending radiation to the convex corners until the incidence angleof the incoming radiation is close to the acceptance angle limit. Inthis way, the losses due to the rounding of the convex corners do notaffect the rays arriving perpendicular (or close to it) to the array.FIG. 25 illustrates how the single sheet of lenses forms a Kohlerintegrator by depicting in phantom lines the grooves and lenses, as theyappear as virtual images in the reflecting groove surfaces. The lightrays are correspondingly depicted as straight rays crossing thereflective surface into the virtual realm.

7. Free-Form Grooves with Non-Flat Cross Section

Let us consider now the design of a grooved parabolic reflector similarto the one described in FIGS. 10 a and 10 b but using large non-flatcross sectional groove profiles. The reasoning and analysis providedabove can be easily extrapolated for the design of other generalreflectors.

FIG. 26 shows such a design with 10 free-form grooves. This number ofgrooves is one with which, for a material with n=1.5, all the designrays undergo TIR. Other designs, such as those composed by 4 and 8reflectors, need metallization if efficiency is to be optimized, but arealso interesting for square emitters, because the guiding lines can alllie on planes of symmetry of a square, and thus the symmetry conditionindicated in FIG. 9 is fulfilled. Such symmetry allows the emissionintensity to remain approximately square. Of course, the same appliesfor a square receiver and its angular sensitivity.

The groove design can be done with two methods: a first one in which thesolution is approximated as a sequence of 2D designs (preferably SMS2Ddesigns, as described in the previous sections) along the groove, andthe second one in which the calculation is done directly in 3D with theSMS3D method described in U.S. Pat. No. 7,460,985 to Benitez et al.

7.1 Sequence of 2D Designs Along the Groove

Referring to FIG. 27, consider that in this example all the rays offocal point 270 are going to be collimated parallel to direction vector271, which will be equivalent to a focal point located at infinity.Consider the parabola 272 as guiding curve for the groove. Without lossof generality, consider that parabola 272 is located in meridian planey=0 and x>0, so its equation can be written in polar coordinates asr(θ)=2f/(1+cos(θ)), where θ is angle 274 a and r is distance 274 b. Forevery value of the parameter θ, we can calculate the plane perpendicularto the parabola 272 at the point 273 defined by (θ, r(θ)). The localaxes of that perpendicular plane are y′−z′. We can compute theprojection of point 270 and direction vector 271 on plane 274 whichgives point 270 p, and direction vector 271 p, respectively. Thedistance between points 273 and 270 p is a(θ)=r(θ)cos(θ/2).

The intersection of the groove with the plane y′−z′ is then designedinside that plane as a 2D design to focus point 270 p and directionvector 271 p. If the groove has equation z′=f(y′, a(θ)), then thefree-form surface equation of the groove for y>0, x>0 lying on guidingcurve 272 a can be expressed as:

$\begin{matrix}{{{x\left( {\theta,y^{\prime}} \right)} = {{{r(\theta)}{\sin(\theta)}} - {{f\left( {y^{\prime},{a(\theta)}} \right)}{\sin\left( \frac{\theta}{2} \right)}}}}{{y\left( {\theta,y^{\prime}} \right)} = y^{\prime}}{{z\left( {\theta,y^{\prime}} \right)} = {f - {{r(\theta)}{\cos(\theta)}} + {{f\left( {y^{\prime},{a(\theta)}} \right)}{\cos\left( \frac{\theta}{2} \right)}}}}} & (17)\end{matrix}$

The cross sectional 2D designs are the parabolic approximations given byequation (2), the free-form equation of the groove facet in the x>0, y>0region is given by:

$\begin{matrix}{{{x\left( {\theta,y^{\prime}} \right)} = {{{r(\theta)}{\sin(\theta)}} - {\left( {y^{\prime} + {\frac{1}{4{a(\theta)}}y^{\prime 2}}} \right){\sin\left( \frac{\theta}{2} \right)}}}}{{y\left( {\theta,y^{\prime}} \right)} = {{y^{\prime}{z\left( {\theta,y^{\prime}} \right)}} = {f - {{r(\theta)}{\cos(\theta)}} + {\left( {y^{\prime} + {\frac{1}{4{a(\theta)}}y^{\prime 2}}} \right){\cos\left( \frac{\theta}{2} \right)}}}}}} & (18)\end{matrix}$

Substituting a(θ) and r(θ):

$\begin{matrix}{{{x\left( {\theta,y^{\prime}} \right)} = {{\frac{2f}{1 + {\cos(\theta)}}{\sin(\theta)}} - {\left( {y^{\prime} + {\frac{\cos\left( \frac{\theta}{2} \right)}{4f}y^{\prime 2}}} \right){\sin\left( \frac{\theta}{2} \right)}}}}{{y\left( {\theta,y^{\prime}} \right)} = {{y^{\prime}{z\left( {\theta,y^{\prime}} \right)}} = {f - {\frac{2f}{1 + {\cos(\theta)}}{\cos(\theta)}} + {\left( {y^{\prime} + {\frac{\cos\left( \frac{\theta}{2} \right)}{4f}y^{\prime 2}}} \right){\cos\left( \frac{\theta}{2} \right)}}}}}} & (19)\end{matrix}$

The rest of facets are obtained by applying the corresponding symmetries(dividing the circle into 20 sectors with a plane of reflection everyπ/10 radians for the 10 groove design).

If, instead of using the approximate 2D design of equation (2), one ofthe exact SMS 2D designs of section 5 can be used, the free-form surfacewill perform more accurately. The expression of z′=f(y′, a(θ)) will nothave, in general, a close form in that case.

In the neighborhood of the parabola 272 any of the solutions is a goodapproximation, and this design converges to that of FIG. 10 a and FIG.10 b when the number of grooves becomes sufficiently large.

7.2 Free-Form SMS 3D Designs

Referring to FIG. 28, the exact solution of the groove is done with theSMS3D (simultaneous multiple surface in 3 dimensions) design methodfollowing. The construction is analogous to the SMS2D designs of Type Idescribed in section 5 above, but with the points in three dimensions.The initial point 151 in FIG. 15 is now a point 281 of a parabolicguiding line in FIG. 28. The sequence of calculated points 151 a to 151d is the sequence 281 a to 281 d. The curves obtained are contained inthe free-form groove surfaces, and selecting as many initial points 281as desired along the guiding line provides as many points as desired onthe surfaces. The curves obtained are in general non planar (i.e. theirtorsion is non-zero), except in the case of the initial point at x=0(for which symmetry reduces the problem to the SMS2D design in FIG. 15).Note that, analogously to the fact that the straight line joining 151and 152 does not bisect the 90° corner at 151 in FIG. 15, the SMS 3Dproblem has the degree of freedom of the selections of the orientationof the corners at the points 281 along the guiding line.

8. RXI with Grooved Reflectors

FIG. 29 shows a cross section view of a rotational-symmetric air-gapRXIR (Refraction-refleXion-Internal reflection-Refraction) device 301,described in U.S. Pat. No. 6,896,381. Its 3 optical surfaces are theaperture 302, the reflector 303, and the dome 304. The aperture 302 actsboth as a refracting surface and as a totally internally reflectingsurface. The device may be classed as an RXI device if the surface ofthe dome is shaped so that refraction at that surface does notmaterially affect the optical properties of the device. The device 301may be described as an RIXR or IXR device (reversing the order of theoptical surfaces) if use as a source collimator, rather than aconcentrating collector, is considered primary.

FIG. 30 shows the aperture 302 and FIG. 31 shows reflector surface 303and the cavity surface 304. The reflector surface 303 is usually coveredby a metallic reflector. Its 3 optical surfaces are the entry aperture302, the reflector 303 (cross-hatched in FIG. 31), and the dome 304(dotted). When used as a collimating element, the source (an LED forinstance) is placed inside the dome. The light emitted by the source isrefracted at the dome surface 304 and sent towards the aperture 302where the light is reflected. This reflection is due to Total InternalReflection (TIR), and occurs for all the points of the apertureexcepting for the points of an inner circle 305 (shaded in FIG. 30)where the angle with which the light impinges on the aperture 302 is toosteep for TIR. In general this circle is covered by a metal layer(aluminum or silver, for instance) to get a metallic reflector. Thelight reflected by the aperture is sent towards the reflector 303 whereit is reflected again.

The light reflected by the reflector 303 is sent back to the aperture302. Unlike the first pass by the aperture, now the rays form an anglewith the normals to the aperture surface such that they are refractedand exit the RXI. A proper design of the three optical surfaces gets ahigh collimation from a wide spreading source such as the LED and in avery compact device.

A substantial part of the cost of the RXI is due to the need for themetal covers or coatings on surfaces 303 and 305. These metal reflectors(both or only one) can be substituted by a grooved reflector working byTIR. This allows the metallization process to be eliminated.

FIG. 32 shows a cross-section view of a rotational-symmetric air-gap RXI321 similar to that in FIG. 31, but wherein the reflector surface hasbeen substituted by a V-shaped grooved reflector 322.

FIG. 33 shows the three optical surfaces of an RXI 331 in which themetallic surfaces (reflectors 303 and 305 in FIG. 29) have beensubstituted by grooved reflectors 333 and 335. The dome 334 and thepoints of the aperture 332 not belonging to the inner circle 335 remainwith the same shape as in a metalized device. This RXI does not need anymetallization.

FIG. 34 a shows the reflector surface 333 of device 331. Since the dome334 is at the center of the RXI, this grooved reflector 333 does notcontain any point of the axis. This is an advantage for the groovedreflector because the density of the grooves theoretically increases toinfinity when approaching the axis, and that means that the losses dueto rounding of the corners of the grooves will also increase towards theaxis.

That advantage is not present for the reflector 335 at the center of theaperture 332, seen in FIG. 34 b, which does extend to the axis.

This is more clearly seen in the close up of the reflector 335 shown inFIG. 35.

The losses due to corner rounding are limited by the manufacturingprocess, and cannot be decreased except by reducing the total length ofthe corners. For this reason it is preferred to start designing an RXIso it does not need the inner circle reflector 305, i.e., such that allthe radiation of interest undergoes TIR after refraction at the dome304. This can be achieved by two means:

One way of designing an RXI is by prescribing the dome and thendesigning the aperture and the reflector. The dome can be prescribedwith a cusp such that it refracts light. FIG. 36 shows the cross-sectionof one of these RXI 361. The dome 364 has a central cusp 365 instead ofthe rounded top shown in FIGS. 29 and 32. Thus, almost all the lightemerging from the dome is refracted away from the center of the aperture362 at angles at which it can be internally reflected by the aperture362 to reach the reflector. In this case, only the reflector surfacemust be replaced by a grooved reflector 363 to get a metal-less RXI.

The second way is to replace the reflector 305 by a lens. This solutionsmoothes the collimation because the lens is not able to get as goodcollimation as the RXI does, but this is not necessarily a disadvantagefor some applications. FIG. 37 shows this solution. This RXI 371 isformed by a single dielectric piece and does not need any metallizationto work properly. The center of the dome 374 and the inner circle of theaperture 372 have been replaced by two refractive surfaces forming alens 375. The reflector surface 373 is a grooved reflector.

8.1 RXI with Conical Reflector

The reflector surfaces 363 and 373 of FIG. 36 and FIG. 37 look almostconical (i.e., the cross section of these surfaces looks like twosymmetric straight lines that meet at a point of the axis of symmetry.The design of the RXI's reflector surface can be forced to be conical.This is achieved by prescribing (as conical) the reflector surface anddesigning the cavity and the entry aperture surface, instead of what isdone usually which is prescribing the cavity surface and designing thereflector and the entry aperture surface. The resulting RXI design isvery close to that of FIG. 36 and FIG. 37 because those RXIs have analmost conical reflector surface. Since the cone is a developablesurface, this allows manufacturing the reflector by cutting out a sectorof circular ring from a flat reflector film. This reflector piece isadapted to the RXI reflector surface. The reflector piece can bemechanically fixed to the dielectric body of the RXI by gluing (orcoinjecting RXI and reflector film) although a higher reflectivity canbe achieved with a small air gap between the RXI dielectric bulk and thereflector film. The film can be a mirror reflection film (such as ECP305 film or another mirror reflector Vikuiti film from 3M) or adielectric retro-reflector film similar to the Vikuiti™ BrightnessEnhancement Films (BEF) (also from 3M) but in which the grooves haveradial symmetry instead of the parallel, straight linear pattern of theBEF. FIG. 38 shows the piece of flat radial grooved film 381 oncetrimmed.

8.2 Grooved Reflectors Used as Intensity and Irradiance Mixers

In a conventional RXI used as a collimator of a wide angle source likean LED chip, there is an approximate correspondence between theirradiance at the emitting surface and the intensity in the far field.If there is an obstacle in the emitting surface (for instance a metalcontact) the far field has a corresponding dark region. If we use 4different color LEDs instead of a single chip LED, then the far fieldpattern is of different colors in 4 different sectors. Becausereflection in a grooved reflector is not a conventional reflection asexplained in Section 0, there is not the same correspondence between theirradiance at the source emitter and the far field intensity. Thiseffect can be used to mix the light. For instance we can use 4 differentcolor LEDs chips at the emitter location and get a blended color in thefar field (and also near field) without the need of any extra element.For this purpose it is better to design a conventional RXI close to theaplanatic condition.

FIG. 39 a shows the intensity pattern for a conventional RXI close tothe aplanatic condition. The vertical axis is the intensity in candlesper lumen emitted by the emitter. The two horizontal axes are angularcoordinates of the direction in degrees and centered at the directionnormal to the RXI. The emitter size is 1×1 in arbitrary units and theaperture diameter is 35 in these units. FIG. 39 b shows the intensitypattern when the emitter is placed off the axis so its center is at(0.6,0.6). Since the emitter size is 1×1, no point of the emitter is onthe axis. The result of this change in the intensity is essentially thatthe pattern is shifted from the center. If, in addition to this off-axisemitter shifting, the reflector surface of the RXI is substituted by asuitably designed grooved reflector then the intensity pattern regainsrotational symmetry around the normal direction, even though the emitteris completely off-axis. This new intensity pattern is shown in FIG. 40.

This figure also shows a decrease of the intensity due to the averagingeffect of the intensity pattern of FIG. 39 b around the origin.

The important fact here is that the intensity pattern is almostrotationally symmetric about the RXI axis, even though the emission isnot. This effect can be used to mix light of different colors by placing4 different color LED chips so the axis of rotational symmetry of theoriginal RXI passes through the center of this arrangement. Each LEDproduces a rotationally symmetric pattern which is mixed with thepatterns of the other three LEDs.

The same effect can be used to homogenize the irradiance pattern at thereceiver when the RXI is used as a concentrator of radiation, forinstance for photovoltaic solar energy applications where the receiveris a solar cell. In a conventional RXI there is a hot spot on the cellat a position corresponding to the sun's angular position with respectto the concentrator. The use of a grooved reflector can make theirradiance at the cell more uniform without degrading the acceptanceangle or the efficiency.

9. Cavities with Grooved Reflectors

FIG. 41 shows a reflecting cavity 411 as disclosed in US patentApplication 2008/08291682 A1 by Falicoff et al., the purpose of which isto increase the brightness emitted by an LED chip 412. The cavitycomprises a lens 413 and an elliptic reflector 414. Part of theradiation 415 emitted by the chip finds the lens and is collimated byit. The remaining radiation 416 is reflected back to the chip,contributing to an increase in the brightness of radiation 417 exitingthe device. The lens 413 is not necessary for increasing the brightness.

The reflector surface in this device can be replaced by a groovedreflector as shown in the cavity 421 of FIG. 42. Unlike the cavity ofFIG. 41, in the case of using the grooved reflector of FIG. 42 it ispreferable to have a circular emitting surface instead of the squaredone 422. The lens 423 is identical to the previous one. The only changeis in the grooved reflector 424.

When the size of the emitter is small, the configuration shown in FIG.43 may be more convenient. In this case the whole optical device 431 ismade of a single dielectric piece of refractive index n. The LED chip432 is not in optical contact with this piece, i.e., there is a smallair gap between them at the device entrance 435. The light entering intothe dielectric is collimated within the angle arcsin(1/n). Part of thislight reaches the lens 433 and exits the device with a certaincollimation. The remaining part of this light is reflected back by thegrooved reflector 434 which does not need to cover a hemisphere becauseof the partial collimation by refraction at the entrance 435. TheFresnel reflection at the flat surface 435 helps to increase thebrightness of the source.

FIG. 44 and FIG. 45 show another possibility to provide such abrightness enhancement and collimating cavity, using a groove profile asthat of FIG. 15. The device 440 shown in FIGS. 44 and 45 has anapproximately cylindrical side wall that does not need to be reflectedbecause of partial collimation by refraction at the entrance, as in FIG.43. On the other hand, FIG. 46 shows the use of a single groove 462 toprovide such a brightness enhancement and collimating cavity.

1. A method of designing a grooved reflector, comprising: selecting twogiven wavefronts; and designing two surfaces meeting at an edge to forma groove such that the rays of each of the given wavefront become raysof a respective one of the given wavefronts after a reflection at eachof the surfaces.
 2. The method according to claim 1, wherein the rays ofeach given wavefront become rays of the other given wavefront after saidreflections.
 3. The method according to claim 1, wherein the rays ofeach given wavefront become rays of the same given wavefront after saidreflections.
 4. The method according to claim 1, wherein the givenwavefronts are curved.
 5. The method according to claim 1, furthercomprising selecting a point such that the two rays associated with thetwo given wavefronts and passing through that point are not coincident,and designing the two said surfaces such that said given point is onsaid edge at which the two surfaces meet.
 6. The method according toclaim 1, comprising designing an initial portion of each said surfacesuch that a fan of incident rays reflected at an edge point of the otherof said surfaces are collimated by said initial portion into rays of oneof said given wavefronts.
 7. The method according to claim 6, furthercomprising designing a subsequent portion of each said surface such thatrays of rays of one of said given wavefronts become rays of the other ofsaid given wavefronts after reflection at said initial portion of one ofsaid surfaces and said subsequent portion of the other of said surfaces,in either order.
 8. The method of claim 1, comprising designing aguiding line, along which said edge of the groove lies, such that theguiding line lies on a surface which coincides with a Cartesian ovalreflector that would couple said two given wavefronts.
 9. The method ofclaim 8, wherein the reflector has a plurality of grooves, the saidguiding lines of which lie on the same said Cartesian oval surface. 10.The method of claim 1, wherein for every point of said edge the tangentvector to said edge is coplanar to the rays of the two wavefrontspassing through said point.
 11. The method of claim 1, wherein at saidedge the normal vectors to the two surfaces forming a groove reflectorsform 90° to each other.
 12. The method of claim 1, further comprisingtruncating polynomial representations of the two designed surfaces. 13.The method of claim 1, further comprising forming said surfaces asKohler integrators.
 14. The method of claim 1, further comprisingdesigning the two surfaces as surfaces of a dielectric body throughwhich the rays pass.
 15. The method of claim 14, further comprising thetwo surfaces as totally internally reflecting surfaces of the dielectricbody.
 16. The method of claim 14, further comprising designing anopposite surface of said dielectric body with refracting elementsthrough which said rays of said wavefronts enter and leave saiddielectric body.
 17. The method of claim 16, further comprisingdesigning said refracting elements to form Kohler pairs with theirimages after reflection in both said surfaces.
 18. The method of claim1, further comprising manufacturing a grooved reflector according to thedesign.
 19. The method of claim 1, further comprising manufacturing anoptical device incorporating said grooved reflector.